How to teach and think about spontaneous wave function collapse theories: not like before
Lajos Di´osi Wigner Research Center for Physics H-1525 Budapest 114. P.O.Box 49, Hungary (Dated: October 28, 2017)
A simple and natural introduction to the concept and formalism of spontaneous wave function collapse can and should be based on textbook knowledge of standard quantum state collapse and monitoring. This approach explains the origin of noise driving the paradigmatic stochastic Schr¨odinger equations of spontaneous localiza- tion of the wave function Ψ. It reveals, on the other hand, that these equations are empirically redundant and the master equations of the noise-averaged stateρ ˆ are the only empirically testable dynamics in current spontaneous collapse theories.
I. INTRODUCTION
“We are being captured in the old castle of standard quantum mechanics. Sometimes we think that we have walked into a new wing. It belongs to the old one, however” [1].
Year 1986 marked the birth of two theories, prototypes of what we call theory of spon- taneous wave function collapse. Both the GRW paper published in Physical Review D [3] followed by Bell’s insightful work [4] and the author’s Thesis [5] constructed strict stochastic jump equations to explain unconditional emergence of classical behavior in large quantum systems. Subsequently, both theories obtained their time-continuous versions, driven by white-noise rather than by stochastic jumps. The corresponding refinement of the GRW pro- posal [6–8] is the Continuous Spontaneous Localization (CSL) theory, the author’s gravity- related spontaneous collapse theory [5, 9, 10] used to be called DP theory after Penrose concluded to the same equation for the characteristic time of spontaneous collapse in large bodies [11]. These theories modified the standard theory of quantum mechanics in order to describe the irreversible process of wave function collapse. The mathematical structure of modification surprised the proponents themselves and it looked strange and original for 2 many of the interested as well. In fact, these theories were considered new physics with new mathematical structures, to replace standard equations like Schr¨odinger’s. The predicted effects of spontaneous collapses are extreme small and have thus remained untestable for the lack of experimental technique. After three decades, fortunately, tests on nanomasses are now becoming gradually available. Theories like GRW, CSL, DP have not changed over the decades apart from their parameter ambiguities, see reviews by Bassi et al. [12, 13]. But our understanding and teaching spontaneous collapse should be revised radically. Personally, I knew that GRW’s random jumps looked like unsharp measurements but, in the late nineteen-eighty’s, I believed that unsharp measurements were phenomenologi- cal modifications of von Neumann standard ones. My belief extended also for the time- continuous limit of unsharp measurements [14] that DP collapse equations [10] were based on. Finally in the nineteen-nineties I got rid of my ignorance and learned that unsharp measurements and my time-continuous measurement (monitoring) could have equally been derived from standard quantum theory [15, 16]. That was disappointing [1]. Excitement about the radical novelty of our modified quan- tum mechanics evaporated. Novelty got reduced to the concept that tiny collapses, that get amplified for bulk degrees of freedom, happen everywhere and without measurement devices. That’s why we call them spontaneous. But they are standard collapses otherwise. I have accordingly stressed upon their revised interpretation recently [2], the present work is arguing further toward such demand.
II. HOW TO TEACH GRW SPONTANEOUS COLLAPSE?
We should build as much as possible on standard knowledge, using standard concepts, equations, terminology. Key notion is unsharp generalized measurement, which has been standard ever since von Neumann showed how inserting an ancilla between object and mea- suring device will control measurement unsharpness [17]. Hence we are in the best pedagog- ical position to explain GRW theory to educated physicists. No doubt, for old generations measurement means the projective (sharp) one but this has changed recently due to the boom in quantum information science. For younger scientists, generalized measurements are the standard ones, projective measurements are the specific case [18, 19]. For new gen- eration, there is a natural way to get acquainted with spontaneous collapse. The correct 3
and efficient teaching goes like this. GRW theory assumes that independent position measurements of unsharpness (precision) √ −5 rC / 2, with GRW choice rC = 10 cm, are happening randomly at average frequency −19 λ = 10 Hz on each (non-relativistic) particle in the Universe. The two parameters rC , λ are considered new universal constants of Nature. The mathematical model of unsharp measurements is exactly the same as for independent von Neumann detectors [17] where √ the Gaussian ancilla wave function has the width σ = rC / 2 [1]. The difference from standard von Neumann detection is the concept of being spontaneous: GRW measurements are supposed to happen without presence of detectors. The merit of GRW is wave function localization in bulk degrees of freedom like, e.g., the center-of-mass (c.o.m.) of large objects. Quantum theory allows for arbitrary large quantum fluctuations of macroscopic degrees of freedom in large quantized systems. The extreme ex- ample is a Schr¨odingercat state where two macroscopically different wave functions would be superposed. In GRW theory such macroscopic superpositions or fluctuations become suppressed by GRW spontaneous measurements but the superpositions of microscopic de- grees of freedom will invariably survive. These complementary features are guaranteed by the chosen values of parameters σ and λ. Due to the extreme low rate of measurements, individual particles are almost never measured. But among an Avogadro number (A) of constituents some N = Aλ ∼ 104 become spontaneously measured in each second, meaning that their collective variables, e.g.: center-of-mass, become measured each second at preci- √ sion σ/ N ∼ 10−7 cm, leading to extreme sharp c.o.m. localization on the long run. That’s what we expect of spontaneous localization theories. The mathematical model is the following. We model the Universe or part of it by a quantized N-body system satisfying the Schr¨odingerequation
d |Ψi i = − Hˆ |Ψi (1) dt ~ apart from instances of spontaneous position measurements that happen randomly and inde- pendently at rate λ on every constituent. Spontaneous position measurements are standard
generalized measurements. Accordingly, when the k’th coordinate xˆk endures a measure- ment, the quantum state undergoes the following collapse: p G(xk − xˆk) |Ψi |Ψi =⇒ p . (2) k G(xk − xˆk) |Ψi k 4
The effects of unsharp position measurement take the Gaussian form:
1 (x − xˆ)2 G(x − xˆ ) = exp − , (3) k k (2πσ2)3/2 2σ2
where xk is the random outcome of the unsharp position measurement on xˆk, and σ sets the scale of unsharpness (precision). The probability of the outcomes xk is defined by the standard rule: p 2 p(xk) = k G(xk − xˆk) |Ψi k . (4)
We have thus specified the mathematical model of GRW in terms of standard unsharp position measurements targeting every constituent at rate λ and precision σ. These mea- surements are selective measurements if we assume that the measurement outcomes xk are accessible. If they are not, we talk about non-selective measurements and the jump equation (2) should be averaged over the outcomes, according to the probability distribution (4). The mathematical model of the GRW theory reduces to the following master equation for the density matrixρ ˆ:
dρˆ i X Z = − [H,ˆ ρˆ] + λ dx pG(x − xˆ )ˆρpG(x − xˆ ) − λρˆ dt k k k k k ~ k i ˆ X = − [H, ρˆ] + λ D[xˆk]ˆρ. (5) ~ k The decoherence superoperator is defined by Z D[xˆ]ˆρ = dxpG(x − xˆ)ˆρpG(x − xˆ) − ρ.ˆ (6)
We can analytically calculate it in coordinate representation ρ(x, x0) of the density matrix. Its contribution on the rhs of the master equation (5) shows spatial decoherence, saturating for large separations:
0 0 2 dρ(x, x ) X (xk − x ) = ... − λ 1 − exp − k ρ(x, x0), (7) dt 8σ2 k where ellipsis stands for the Hamiltonian part. The amplification mechanism is best illustrated in c.o.m. dynamics. As we said, for the individual particles the decoherence term remains negligible whereas for bulk degrees of freedom, e.g.: the c.o.m., it becomes crucial to damp Schr¨odingercats, as we desired. Assume, for simplicity, free spatial motion of a many-body object. Then the non-selective 5
GRW equation (5) yields the following autonomous equation for the reduced c.o.m. density matrixρ ˆcm: dρˆcm i ˆ = − [Hcm, ρˆcm] + NλD[xˆcm]ˆρcm. (8) dt ~ As we see, the decoherence term concerning the c.o.m. coordinate has been amplified by the number N of the constituents [3] ensuring the desired fast decay of macroscopic superposi- tions:
dρ (x , x0 ) (x − x0 )2 cm cm cm = ... − Nλ 1 − exp − cm cm ρ(x , x0 ). (9) dt 8σ2 cm cm
In the selective evolution the individual GRW measurements (2) entangle the c.o.m., rota- tion, and internal degrees of freedom, hence |Ψcmi does not exist in general. It does in a limiting case of rigid many-body motion when the unitary evolution of |Ψcmi is interrupted by spontaneous σ-precision measurements of the c.o.m. coordinatex ˆcm similarly to (2) just the average rate of the measurements becomes Nλ [20] instead of λ.
III. LOCALIZATION IS NOT TESTABLE, BUT DECOHERENCE IS
Standard concept of selective measurement implies that we have access to the measure- ment outcomes, which are the values xk in GRW. If they are accessible variables then the stochastic jump process of the GRW state vector |Ψi is testable otherwise it is not. If not, then the same spontaneous measurement is called non-selective and what is testable is the density operatorρ ˆ. The stochastic jump process (1-4) becomes illusory and the master equation (5) contains the whole GRW physics.
This latter sentence holds in GRW where, as a matter of fact, the xk’s remain unac- cessible. Consider the conservative preparation-detection scenario. Assume we prepared a well-defined pure initial stateρ ˆ0 = |Ψ0i hΨ0| and by time t later we desire to test it for the presence of GRW collapses (2), we perform no test prior to this one. As a matter of fact, the relevant state isρ ˆt, being the solution of the master equation (5) which does not know about GRW collapses but about GRW decoherence. This is equally valid in the particular case of the macroscopic Schr¨odingercat initial state, i.e., a superposition of c.o.m. at two distant locations. The c.o.m. GRW master equation (8) will exhaustively predict the results of all subsequent tests on the c.o.m. (including the results and statistics of possible naked eye observations). 6
Obviously, inference on stochastic collapse assumes our access to the measurement out- comes. In real laboratory quantum measurements it is the detector design and operation that determine if we have full (or partial) access to the measurement outcomes or we have no access at all. In the case of GRW collapse, accessibility of outcomes it is not a matter of postulation. It is useless to postulate that xk’s are accessible without a prescription of how to access them.
IV. DIGRESSION: RANDOM UNITARY PROCESS INDISTINGUISHABLE FROM GRW
Let us consider an alternative to GRW random process where the stochastic non-linear GRW jumps (2) are replaced by the following stochastic unitary jumps:
|Ψi =⇒ eikxˆk |Ψi , (10) corresponding to the transfer of momentum ~k to the k’th constituent. The probability distribution of momentum transfer is universal, independent of the particle and of the state:
1 k2 p(k) = exp − . (11) (2πσ−2)3/2 2σ−2
The decoherence superoperator acts as Z D[xˆ]ˆρ = dkp(k)eikxˆρˆe−ikxˆ − ρ,ˆ (12) which looks completely different from the GRW structure (6) but it coincides with it! Hence the master equation for the Schr¨odingerdynamics (1) with the averaged unitary jumps (10) will be the the master equation (5) derived earlier for the GRW theory. As we argued in Sec. III, the GRW theory can only be tested at the level of the density operator, no experiment could tell us whether the underlying stochastic process of |Ψi was the GRW stochastic localizing process (1-4) or the above stochastic unitary process.
V. HOW TO THINK ABOUT CSL?
We could repeat what we said concerning correct and efficient teaching of GRW in Section II. This time the standard discipline of modern physics, relevant to CSL, is time-continuous 7 quantum measurement (monitoring) which is just the time-continuous limit of unsharp se- quential measurements similar to those underlying GRW in Section II. Quantum monitoring theory was not yet conceived in 1986 (GRW), it was born in 1988, at it became widely known in the nineties, to become the standard theory of quantum monitoring in the laboratory [15, 16]. It played instrumental role for semi-classical gravity’s consistent introduction to spontaneous collapse theories [21, 22]. Below I utilize the summary of standard Markovian quantum monitoring theory from [21]. So, how should we interpret CSL? It derives from GRW. The discrete sequence of spon- taneous unsharp position measurements is replaced by spontaneous monitoring the spatial number distribution of particles [7] (or, in a later version, of the spatial mass distribution of particles [8]). Accordingly, CSL introduces the smeared mass distribution
X nˆ(x) = G(x − xˆk), (13) k where, this time, the width of the Gaussian is rC . Monitoring yields the measured signal in the form
nt(x) = hΨt| nˆ(x) |Ψti + δnt(x), (14) where δnt(x) is the signal white-noise still depending on the spatial resolution/correlation of monitoring. The CSL signal noise is a spatially uncorrelated white-noise:
1 δn (x)δn (y) = δ(x − y)δ(t − s). (15) E t s 4γ
Just like in the case of GRW sequential spontaneous measurements, the conditional quantum state evolves stochastically, this time according to the following stochastic Schr¨odingerequation, driven by the signal noise in the Ito-sense:
d |Ψi i γ Z Z = − Hˆ − dx (ˆn(x) − hnˆ(x)i)2 + 4γ dx (ˆn(x) − hnˆ(x)i) δn(x) |Ψi . (16) dt ~ 2 So far we have introduced the equations of selective spontaneous monitoring, assuming that the signal (14) is accessible, which won’t be the case, similarly to GRW. In non-selective monitoring, the CSL physics reduces to the signal-averaged evolution of the conditional state, i.e., to the CSL master equation:
dρˆ i γ Z = − [H,ˆ ρˆ] − dx[ˆn(x), [ˆn(x), ρˆ]]. (17) dt ~ 2 8
2 3/2 (Note γ = (4πrC ) λ would ensure the coincidence with GRW’s spatial decoherence rate at the single particle level although CSL defined a slightly different γ [7]). The traditional CSL-teaching differs in a single major point: it does not mention
the theory of monitoring. Hence it does not use the notion of signal nt(x), the equa- tion (14) is not part of it. Instead, CSL’s traditional definition postulates the stochastic Schr¨odingerequation:
d |Ψi i γ Z √ Z = − Hˆ − dx (ˆn(x) − hnˆ(x)i)2 + γ dx (ˆn(x) − hnˆ(x)i) w(x) |Ψi , (18) dt ~ 2 √ which would correspond to the replacement δnt(x) = 2 γwt(x) had CSL derived it from our (16). The traditional CSL dynamics is driven by the spatially uncorrelated standard white-noise, satisfying
Eδwt(x)δws(y) = δ(x − y)δ(t − s). (19)
In CSL narrative (e.g.: [13]) the origin of the noise field as well its anti-Hermitian coupling to densityn ˆ(x) are mentioned among theory elements yet to be justified, still without reference to the spontaneous monitoring interpretation available already for long enough time. Needless to repeat arguments from Section III on GRW. All testable predictions follow from the CSL master equation (17), the stochastic Schr¨odingerequation (18) is empirically redundant, collapse in the claimed quantitative sense is an illusion.
VI. FINAL REMARKS
Disregarding that spontaneous collapse theories are rooted in standard quantum mechan- ical collapse theories with hidden detectors has had too many drawbacks. The principal one is the illusion that the quantitative models of spontaneous collapse (localization) in their current forms are relevant empirically like master equations of spon- taneous decoherence are which have already been under empiric tests due to recent break- throughs in technologies. This illusion is surviving despite no proposals have been ever made for a future experiment to test underlying localization effects of |Ψi beyond decoherence ofρ ˆ; all proposals have so far concerned the dynamical features (e.g.: spontaneous decoherence) of the averaged stateρ ˆ. Secondary drawbacks concern illusions that teaching and interpretation of spontaneous collapse necessitate radical departure from standard quantum theory both conceptually and 9 mathematically. This may have kept philosophers excited, may have prevented students of learning the subject faster, physicists of going deeper into their foundational investigations. Physics research will gradually adapt itself to the option that spontaneous collapse fits better to standard quantum knowledge than we thought of it before. Monitoring theory roots were revealed for DP spontaneous collapse from the beginning, and have been detailed and exploited for CSL, too, recently in [21, 22].
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